Groups satisfying Kaplansky's stable finiteness conjecture

0.19 MB·English

by  Federico Berlai

GROUPS SATISFYING KAPLANSKY’S STABLE FINITENESS CONJECTURE
FEDERICOBERLAI
Abstract. We prove that every {finitely generated residually finite}-by-sofic group satisfies
Kaplansky’sdirectandstablefinitenessconjectures withrespecttoallnoetherianrings.
We use this result to provide countably many new examples of finitely presented non-LEA
groups, for which soficity is still undecided, satisfying these two conjectures. Deligne’s famous
5 exampleSp^2n(Z)ofanonresiduallyfinitegroupisamongourexamples,alongwiththefamilies
01 wofhaemreaplg>am2atisedafprereimper,ondu≥ct3sSaLndn(FZr[1i/spa])f∗rFereSgLronu(Zp[1o/fpr]a)naknrd,HfoNrNalelxrte≥ns2i.onsSLn(Z[1/p])∗Fr,
2

n
a
J
1. Introduction
3
1 In [20, pp. 122-123]Irving Kaplansky posed what nowadaysis knownas Kaplansky’s direct finite-
ness conjecture:
]
R Direct finiteness conjecture - fields. Given a field K and a group G, the group ring K[G] is
G directly finite. That is to say, if x,y ∈K[G] are such that xy =1, then yx=1.
.
h OnecouldlookatthematrixringsMat (K[G])andaskwhetherornottheseringsaredirectly
n×n
at finite for all n ∈ N. This is known as Kaplansky’s stable finiteness conjecture. Although it might
m look stronger than the direct finiteness conjecture, they are in fact equivalent [28]. Hence in this
paper, for the sake of simplicity, we restrict our arguments to direct finiteness.
[
Kaplansky himself proved that, given a field K of characteristic zero and a group G, the group
1 ringK[G]isdirectlyfinite. Sincethenprogresshasbeendone,buttheconjectureisstillunresolved.
v
In [1], Ara, O’Meara and Perera proved that D[G] is directly finite whenever G is a residually
3
9 amenablegroupandD isadivisionring. LaterElekandSzabo´generalizedthisresulttothe wider
8 class of all sofic groups [14, Corollary 4.7].
2 Soficgroupswereintroducedin1999byGromov,inanattempttosolveGottschalk’sconjecture
0
in topological dynamics [16]. The class of sofic groups is far from being completely understood,
.
1 and it still puzzles the experts. In particular, it is not yet known if all groups are sofic.
0 Soficity is stable under many group-theoretic operations [8, 9, 26]. At the same time, it is still
5
unclear how this notion behaves under taking group extensions: While it is known that a sofic-
1
by-amenable groupis againsofic [9, Proposition7.5.14],it is still an open problem whether or not
:
v finite-by-sofic, free-by-sofic or sofic-by-sofic groups, among others, are again sofic groups.
i
X Here we use the standard notation of an A-by-B group to denote a group G with a normal
subgroup N EG such that N ∈A and G/N ∈B, where A and B are given classes of groups (e.g.
r
a A being the free groups and B being the sofic groups, in the case of a free-by-sofic group).
Since there is yet no knownexample of a groupwhich fails to be sofic, the originalKaplansky’s
conjecture and the following variant are still open problems:
Direct finiteness conjecture - division rings. Given a division ring D and a group G, the
group ring D[G] is directly finite.
A majorrecentbreakthroughhas beenmade by Virili[28]. He provedthatany crossedproduct
N ∗G is directly finite whenever N is a left-noetherian ring (respectively: right-noetherian ring)
and G is a sofic group (see Section 3 for the definition and [25, 28] for more details about crossed
products).
2010 Mathematics Subject Classification. 20C07,20E06,20E26, 20F65.
Key words and phrases. Kaplansky’s stable finiteness conjecture, sofic group, Deligne’sgroup, space of marked
groups.
Theauthor issupportedbytheEuropeanResearchCouncil(ERC)grantofProf. GoulnaraArzhantseva, grant
agreementno.259527.
1
2 FEDERICOBERLAI
Group rings are basic examples of crossed products, hence we state another generalization of
the original conjecture:
Direct finiteness conjecture - noetherian rings. Given a noetherian1 ring N and a group G,
the group ring N[G] is directly finite.
As a consequence of his general result on crossed products, Virili deduced that the group ring
N[G] of a {polycyclic-by-finite}-by-sofic group G is directly finite with respect to all noetherian
ringsN [28,Corollary5.4],andthatthegroupringD[G]ofafree-by-soficgroupGisdirectlyfinite
with respect to all division rings D [28, Corollary 5.5]. As mentioned above, the interest in these
classes of groups arises because they are not known to be sofic.
The aim of this paper is to prove the following Theorem, which establishes Kaplansky’s direct
andstable finiteness conjectures for the groupring ofmany groups thatare not knownto be sofic.
Main Theorem. Let N be a noetherian ring and G be a {finitely generated residually finite}-by-
sofic group. Then N[G] is directly finite and, equivalently, stably finite.
As a corollary,we partially extend [28, Corollary 5.5] from division rings to noetherian rings.
CorollaryA. LetN beanoetherian ringandGbea{finitelygeneratedfree}-by-soficgroup. Then
N[G] is directly finite and, equivalently, stably finite.
Moreover,we construct countably many pairwise non-isomorphicfinitely presented groups that
satisfiesbothoftheseconjectures,andthatarenotknowntobesofic. Inparticular,thesegroupsare
notlocallyembeddableintoamenablegroups,alsocalledLEAgroups(seeSection4fordefinitions).
Corollary B. There exists an infinite family {Gi}i∈N of pairwise non-isomorphic finitely pre-
sented non-LEA groups. These groups are not known to be sofic, and D[G ] is directly finite and,
i
equivalently, stably finite, with respect to all division rings D.
The groups described in Corollary B are given by the HNN extensions SLn(Z[1/p])∗Fr and by
the amalgamated free products SLn(Z[1/p])∗Fr SLn(Z[1/p]). Here n ≥ 3 is an integer, p > 2 is a
prime number and F is a free subgroup of SL (Z[1/p]) of rank r, for all r ≥2. See Corollary 4.2
r n
and Corollary 4.3 for the precise statements and the proofs.
The paper is organized as follows: In Section 2 we define the space of marked groups, we recall
some useful properties and we prove preliminary results that will lead to the proof of the Main
Theorem. In Section 3 we prove the Main Theorem and we give some corollaries. In Section 4 we
apply our result to countably many pairwise non-isomorphic groups, which are not yet known to
be sofic, and for which we establish Kaplansky’sdirect finiteness and stable finiteness conjectures.
Deligne’s famous example Sp^(Z) of a non residually finite group is among our interests.
2n
Acknowledgments. The author is very grateful to his advisor,Goulnara Arzhantseva, for inspiring
questionsandsuggestions. HewantstothankSimoneViriliforsharingthe textofhisPhDThesis,
his work[28]andfor manyconstructivediscussionsoverthe topic. Thanks arealsodue to Nikolay
Nikolov, for explaining the proof of [21, Theorem 1].
2. The space of marked groups
In this section, we briefly discuss the space of marked groups. For more details and properties,
see [10, 11, 15]. We prove the following theorem:
Theorem 2.1. Let Qbe agroupand Gbeafinitelygenerated {finitelygenerated residually finite}-
by-Q group. Then G is the limit, in the space of marked groups, of finite-by-Q groups.
We stress the following particular case:
Corollary 2.2. Let G be a finitely generated {finitely generated residually finite}-by-sofic group.
Then G is the limit, in the space of marked groups, of finite-by-sofic groups.
1Themainconcernofthisworkaregroups,henceinwhatfollowswefocusonnoetherianrings,thatis,ringsthat
arebothleft-andright-noetherian,ratherthanspecifyingiftheringisleft-orright-noetherian. Allourstatements
remaintruewhenonerestrictstoleft-noetherian,orright-noetherian,rings.
GROUPS SATISFYING KAPLANSKY’S STABLE FINITENESS CONJECTURE 3
Amarked group isapair(G,S),whereGisafinitely generatedgroupandS isafinite sequence
of elements that generate G. If |S| = n then (G,S) is called an n-marked group. Two n-marked
groups (G,(s ,...,s )) and (G′,(s′,...,s′ )) are isomorphic if the bijection ϕ(s ) := s′ extends
1 n 1 n i i
to an isomorphism of groups ϕ: G→G′. In particular, two marked groups with given generating
setsofdifferentsize areneverisomorphicasmarkedgroups,althoughthey mightbe isomorphicas
abstract groups.
Let G denote the set of n-marked groups, up to isomorphism of marked groups. Then G
n n
corresponds bijectively to the set of normal subgroups of the free group F on n free generators.
n
Let (G,S),(G′,S′) ∈ G and let N, N′ be the normal subgroups of F such that F /N ∼= G,
n n n
Fn/N′ ∼= G′. Let BFn(r) denote the ball of radius r in Fn centered at the identity element. The
function
(1) v(N,N′):=sup{r∈N|N ∩BF (r)=N′∩BF (r)}
n n
defines on G the ultrametric
n
(2) d (G,S),(G′,S′) :=2−v(N,N′).
(cid:0) (cid:1)
Ann-markedgroup(G,S)canbe viewedasan(n+1)-markedgroupbyaddingthetrivialelement
e to S. This defines an isometric embedding of G into G .
G n n+1
LetG := G bethespace of marked groups. Theultrametricsof (2)canbeextendedtoan
ultrametric Sd:nG∈N×nG → R , and G,d is a compact totally disconnected ultrametric space [11].
≥0
(cid:0) (cid:1)
The following fact is well known.
Lemma 2.3. Let (G,S) and {(Gr,Sr)}r∈N be marked groups in Gn and let N,{Nr}r∈N be normal
subgroups of F such that F /N ∼=G and F /N ∼=G . The following are equivalent:
n n n r r
(1) the sequence {(Gr,Sr)}r∈N converges to the point (G,S) in the space Gn,d ;
(cid:0) (cid:1)
(2) for all x ∈ N (respectively: for all y ∈/ N) there exists r¯ such that x ∈ N (respectively:
r
y ∈/ N ) for r ≥r¯;
r
(3) for all R ∈ N there exists r¯ such that the Cayley graphs Cay(G,S) and Cay(G ,S ) have
r r
balls of radius R isomorphic as labeled directed graphs, for all r ≥r¯.
The following easy lemma is used in the proof of Theorem 2.1.
Lemma 2.4. Let K ✂N ✂G be a subnormal chain of groups, where N is finitely generated, and
K has finite index in N.
Then there exists a normal subgroup K✂G, contained in K, such that N/K is finite.
e e
Proof. Since N isfinitely generated,foranygivenpositiveintegerd∈N, N hasonlyfinitely many
subgroups of index d. Let K be the intersection of all subgroups H ✂N with [N : H]= [N : K].
Then K is a finite index chaeracteristic subgroup of N, and hence it is normal in G.
e (cid:3)
We are now ready to prove Theorem 2.1.
Proof of Theorem 2.1. LetGbea{finitelygeneratedresiduallyfinite}-by-Qgroupandsuppose
G is generatedby a finite set S. Let N ✂G be a finitely generated residually finite subgroup such
that G/N ∼=Q.
As N is residually finite, there exists a family {Kr}r∈N of finite index normal subgroups of N
such that K ={e} and K ⊆K for all r∈N. As N is finitely generated,we can assume
that K ✂TGr∈foNr arll r ∈N by Lemr+m1a 2.4.r
r
Consider the family of finite-by-Q groups {G/Kr}r∈N. For every r, let Sr be the image of S in
G/Krunderthecanonicalprojectionλr: G։G/Kr. Weprovethatthesequence{(G/Kr,Sr)}r∈N
converges to (G,S) in G using the second condition of Lemma 2.3.
|S|
Let π: F։G be the canonicalsurjective homomorphismfromthe finitely generatedfree group
F on |S| free generators, and π : F։G/K . Then π =λ ◦π:
r r r r
F❖❖❖❖❖❖π❖r❖π❖❖❖❖❖'' //G(cid:15)(cid:15)λr
G/K
r
4 FEDERICOBERLAI
Set Λ := kerπ and Λ := kerπ . By definition, Λ ≤ Λ for every r, so, to check that the second
r r r
condition of Lemma 2.3 is satisfied, we have only to prove that for all y ∈/ Λ there exists r¯ such
that y ∈/ Λ , for r ≥ r¯. Let y ∈/ Λ and π(y) =: g ∈ G\{e }. Note that if g ∈/ N then g ∈/ K for
r G r
all r, as K ⊆N. Thus, for such g ∈/ N we have
r
(3) π (y)=λ (π(y))=λ (g)6=e ∀r ∈N,
r r r G/Kr
that is to say, y ∈/ Λ for all r∈N.
r
If g ∈N \{e }, then there exists r¯such that g ∈/ K for all r ≥r¯, because K ={e} and
G r Tr∈N r
because this family of normal subgroups is totally ordered by inclusion. In particular,
(4) π (y)=λ (g)6=e ∀r ≥r¯.
r r G/Kr
This implies that y ∈/ Λ for r ≥r¯, and the proof is completed.
r
(cid:3)
3. Proof of the Main Theorem
WefirstshowthattheassertionsofKaplansky’sdirectfinitenessandstablefinitenessconjectures
are preserved under taking limits in the space of marked groups.
Proposition 3.1. Let N be a noetherian ring and (G,S) be the limit, in the space of marked
groups, of the sequence {(Gr,Sr)}r∈N. If N[Gr] is directly finite (respectively: stably finite) for all
r∈N then so is N[G].
Proof. SupposefirstthatN[G ]isdirectlyfiniteforallr ∈N,andconsidertwonon-trivialelements
r
x= k g and y = h g ∈N[G] such that xy =1. We want to prove that yx=1.
Pg∈G g Pg∈G g
Let yx= l g and consider
Pg∈G g
m:=max{kgk |k 6=0 or h 6=0 or l 6=0},
S g g g
where k−k denotes the norm induced by the word metric on G given by the finite generating set
S
S. That is to say:
kgk :=min{k|g =s ...s , s ∈S∪S−1}.
S 1 k i
Since the sequence {(Gr,Sr)}r∈N converges to (G,S), by Lemma 2.3 there exists r¯such that, for
allr≥r¯,Cay(G,S)andCay(G ,S )haveballsofradiusmisomorphicaslabeleddirectedgraphs.
r r
Thegroup-coefficientsofxandy havethe samepartialmultiplicationinGasinG ,andmoreover
r
yx=1 in N[G ]. It thus follows that yx=1 in N[G].
r
The same arguments work when N[G ] is stably finite for all r ∈N.
r
(cid:3)
Proof of the Main Theorem. First we suppose that the group in question is finitely gener-
ated, so let G be a finitely generated {finitely generated residually finite}-by-sofic group. By
Corollary 2.2, G is the limit of finite-by-sofic groups, which satisfy Kaplansky’s direct finiteness
conjecture with respect to all noetherian rings [28, Corollary 5.4]. Hence, by Proposition 3.1, G
satisfies the conjecture with respect to all noetherian rings.
If the group G is not finitely generated, consider a finitely generated residually finite normal
subgroupK suchthatG/K issofic. ThenGisthedirectedunionofitsfinitelygeneratedsubgroups
containing such K:
(5) G= {H |K ≤H ≤G and H is finitely generated}.
[
Thesearefinitely generated{finitely generatedresiduallyfinite}-by-soficgroups,andhence satisfy
Kaplansky’s direct finiteness conjecture by the first part of the proof.
Fix a noetherian ring N and consider two elements x,y ∈ N[G] such that xy = 1. The group-
coefficientsofx, y andyx,beingnon-trivialonlyfinitely manytimes,sitinsomefinitelygenerated
subgroup H ≤G appearing in the directed union in (5).
The group ring N[H] is directly finite by the first part of this proof, so yx = 1 in N[H]. This
implies that yx=1 in N[G]. Thus, G satisfies the conjecture. (cid:3)
GROUPS SATISFYING KAPLANSKY’S STABLE FINITENESS CONJECTURE 5
There is a variant of Lemma 2.4, in the case when the group N/K is solvable: Assume we are
givena subnormalchainK✂N✂Gandsuppose that N/K is solvable,thenthere exists anormal
subgroup K✂G, contained in K, such that N/K is solvable as well.
To the aeuthor’s knowledge, the following are oepen questions:
Question 3.2. Let N be a noetherian ring and G be a solvable-by-sofic group. Is N[G] directly
finite?
Question3.3. LetN beanoetherianringandGbeasolvablegroup. Doesthereexistanoetherian
ring N′ such that N[G] embeds into N′?
An affirmative answer to the latter question implies an affirmative answer to Question 3.2. If
Question 3.2 has an affirmative answer,then our argumentas in the proof of Theorem 2.1 implies
thatafinitely generated{residuallysolvable}-by-soficgroupGis the limitinG ofsolvable-by-sofic
groups, and hence that a {residually solvable}-by-sofic group is directly finite.
We recall now the definition and basic facts on crossed products. They are useful in the appli-
cations of our Main Theorem.
Given a ring R and a group G, a crossed product R∗G of G over R is a ring constructed as
follows. Assign uniquely to every g ∈G a symbol g¯, and let G¯ be the collection of these symbols.
As a set,
R∗G:= r g¯|g ∈G,r ∈R is almost always 0 .
nX g g o
g∈G
The sum is defined component-wise,
r g¯ + s g¯ := (r +s )g¯.
(cid:16)X g (cid:17) (cid:16)X g (cid:17) X g g
g∈G g∈G g∈G
The product in R∗G is specified in terms of two maps
τ: G×G→U(R), σ: G→Aut(R),
where U(R) is the group of units of R and Aut(R) is the group of ring automorphisms of R. Let
rσ(g) denote the result of the action of σ(g) on r. Then, for all r ∈ R and g,g ,g ,g ∈ G, the
1 2 3
maps σ and τ satisfy
σ(e)=1, τ(e,g)=τ(g,e)=1,
and
τ(g ,g )τ(g g ,g )=τ(g ,g )σ(g1)τ(g ,g g ), rσ(g2)σ(g1) =τ(g ,g )rσ(g1g2)τ(g ,g )−1.
1 2 1 2 3 2 3 1 2 3 1 2 1 2
These conditions guaranties that the product
r g¯ · s g¯ := r sσ(h1)τ(h ,h ) g¯
(cid:16)X g (cid:17) (cid:16)X g (cid:17) X(cid:16) X h1 h2 1 2 (cid:17)
g∈G g∈G g∈G h1h2=g
is associative.
Certain crossed products have their own specific name. If the maps σ and τ are trivial, the
crossedproduct R∗G is the groupring R[G]. If σ is trivial, then R∗G=Rt[G] is a twisted group
ring, while if τ is trivial then R∗G=RG is a skew group ring.
Given a normal subgroup N ✂G and a fixed crossed product R∗G, we have
(6) R∗G= R∗N ∗G/N,
(cid:0) (cid:1)
where the latter is some crossed product of the group G/N over the ring R∗N, and R∗N is the
subringofR∗Ginducedbythe subgroupN [25,Lemma1.3](thatis,themapsσ andτ associated
to the crossed product R∗N are the restrictions of the ones associated to R∗G). In particular,
(7) R[G]=R[N]∗G/N.
We now state some interesting corollaries of our main result.
There exist one-relator groups which are not residually finite [3], or not even residually solv-
able [5]. Whether or not all one-relator groups are sofic is a well-known open problem.
6 FEDERICOBERLAI
Wise recently proved that one-relator groups with torsion are residually finite [29], answering
a longstanding conjecture of Baumslag [4]. Combining this deep result of Wise with our Main
Theorem, we obtain:
Corollary 3.4. LetD beadivision ringandGbea {finitelygeneratedone-relator}-by-soficgroup.
Then D[G] is directly finite and, equivalently, stably finite.
Proof. Let N✂G be a normalsubgroupof G suchthat G/N is sofic and N is a finitely generated
one-relatorgroup. IfN istorsion-free,thenitsgroupringD[N]embedsintoadivisionringD′ [22].
By (7) we have that D[G] = D[N]∗G/N. Hence D[G] embeds into D′∗G/N, which is directly
finite [28, Corollary 5.4]. Thus also D[G] is directly finite.
If N has torsion, then it is residually finite [29]. Thus G satisfies the hypotheses of the Main
Theorem, and D[G] is directly finite. (cid:3)
From Corollary 3.4, in the particular case when the sofic group is trivial, we recover the fact
that the group ring of a finitely generated one-relator group is directly and stably finite.
Corollary 3.5. Let D be a division ring and G be a finitely generated one-relator group, then
D[G] is directly finite and, equivalently, stably finite.
Finitely generated right-angled Artin groups are known to be residually finite. We deduce the
following:
Corollary 3.6. Let N be a noetherian ring and G be a {finitely generated right-angled Artin
group}-by-sofic group. Then N[G] is directly finite and, equivalently, stably finite.
Remark 3.7. Here is an alternative way of proving the Main Theorem, which was suggested by
Simone Virili after the first version of this paper was written. In [24] it is observed that, given a
noetherian ring N and
C :={groups G such that N[G] is directly finite},
ifa groupG is fully residuallyC then G∈C. One then arguesthat a {finitely generatedresidually
finite}-by-soficgroupisresiduallyfinite-by-sofic,whichisequivalentofbeingfullyresiduallyfinite-
by-sofic. Hence the Main Theorem follows.
4. Examples
We now apply our Main Theorem to some concrete groups, whose (non-)soficity still intrigues
the experts. We conclude that they satisfy Kaplansky’s direct and stable finiteness conjectures.
Moreover, we provide countably many new explicit presentations of groups satisfying these two
conjectures. These groups are not Locally Embeddable into Amenable (LEA for short), and it is
not known whether or not they are sofic.
Afinitely generatedLEAgroupisthe limitinG ofamenablegroups. Inparticular,LEAgroups
are sofic. There exist examples of sofic groups which are not LEA, and the class of LEA groups is
the biggest known class of groups strictly contained in the class of sofic groups.
In what follows, we use the fact that a finitely presented LEA group is residually amenable [9,
Proposition7.3.8]. We refer to [9, §7.3] for more informations about LEA groups.
4.1. Deligne’s group. A famous example considered by Deligne [12] is the following. Let n ≥ 2
be an integer and Sp^(Z) be the preimage of the symplectic groupSp (Z) in the universalcover
2n 2n
Sp^(R) of Sp (R). It is known [12] that Sp^(Z) is given by the following central extension
2n 2n 2n
(8) {e}−→Z−→Sp^(Z)−→Sp (Z)−→{e}.
2n 2n
The group Sp^(Z) is finitely presented as it is an extension of two finitely presented groups.
2n
Moreover,thegroupisnotresiduallyfinite[12]anditsatisfiesKazhdan’sproperty(T)[6,Example
1.7.13 (iii)]. This immediately implies that it is not an LEA group.
OurMainTheoremimpliesthatN[Sp^(Z)]isdirectlyfiniteforallnoetherianringsN. Indeed,
2n
Sp^(Z) is {finitely generated free}-by-sofic,as shown by (8).
2n
GROUPS SATISFYING KAPLANSKY’S STABLE FINITENESS CONJECTURE 7
4.2. Finitely presented amalgamated products and HNN extensions. From now on, if Γ
is a group then Γ¯ denotes an isomorphic copy of Γ. If Γ = hX | Ri is a presentation of the group
Γ, let X¯ and R¯ denote the same generators and relators in the isomorphic copy Γ¯.
Let p > 2 be a prime number and n ≥ 3. The group SL (Z[1/p]) is finitely presented [17,
n
Theorem4.3.21]andhasKazhdan’sproperty(T)[6]. Moreover,itsatisfiesthecongruencesubgroup
property [2]. This means that every finite index subgroup H ≤SL (Z[1/p]) contains the kernel of
n
the natural projection
(9) π : SL (Z[1/p])։SL (Z/qZ),
q n n
for some q coprime with p. In particular, if the finite index subgroup H is normal in SL (Z[1/p]),
n
it follows that
SL (Z[1/p]) SL (Z[1/p])
(10) n ∼=SL (Z/qZ) or n ∼=PSL (Z/qZ),
n n
H H
for exactly one q coprime with p.
In what follows, given an element x ∈ SL (Z[1/p]) and a projection π from SL (Z[1/p]) onto
n n
SL (Z/qZ) or PSL (Z/qZ), we denote the order of π(x) by o .
n n x
The proof of the following theorem is adapted from [21, Theorem 1], where an analogous fact
is proved, but in the case when the amalgamated subgroup is infinite cyclic. In that case, the
resulting group is known to be sofic. In contrast to [21], our aim is to produce non-LEA groups
thatarenotknowntobesoficandthatsatisfyKaplansky’sdirectandstablefinitenessconjectures.
Theorem 4.1. Let p > 2 be a prime number, n ≥ 3, let Γ := SL (Z[1/p]) = hX | Ri. Let
n
ha,bi=F ≤Γ be the subgroup generated by the matrices
1 2 0 1 0 0
   
a= 0 1 0 , b= 2 1 0 ,
0 0 In−2 0 0 In−2 
where I is the identity matrix of dimension n−2. Then the group
n−2
G:=Γ∗ Γ=hX,X¯ |R,R¯,a=a¯,b=¯bi
F
is not LEA.
Proof. The group G is finitely presented. Hence it is sufficient to prove that it is not residually
amenable. Let
1 2 0
 p 
x= 0 1 0
0 0 I 
n−2
and consider the element g =[x,x¯] ∈G. Since x ∈/ F, using normal forms for the elements of the
amalgamated free product [23, I.11], it follows that g 6=e .
G
Let π: G ։ A be a surjective homomorphism with A amenable. We claim that π(g) = e .
A
Indeed, consider the restriction π ↾ : Γ → π(Γ). The group π(Γ) ≤ A is amenable and moreover
Γ
it is a quotient of Γ, which is a group with Kazhdan’s property (T). Hence π(Γ) is finite and, in
particular, π(x) has finite order o .
x
The element x is unipotent, so π(x) is unipotent too. As the group Γ satisfies the congruence
subgroup property, it follows that π(x) is an element of some SL (Z/qZ) or PSL (Z/qZ), for q
n n
coprime with p. As π(x) is unipotent, the order o divides a power of q. Moreover gcd(p,q) = 1,
x
so gcd(p,o )=1.
x
As xp =a, we have that hπ(a)i≤hπ(x)i and that
o
x
oa =oxp = =ox.
gcd(p,o )
x
Thisimpliesthatthetwofinitegroupshπ(a)iandhπ(x)ihavethesamecardinality,andsohπ(a)i=
hπ(x)i.
The same argument applies to the elements x¯ and a¯, so hπ(a¯)i = hπ(x¯)i. As in the group
G we have a = a¯, it follows that hπ(x)i = hπ(x¯)i, and so π(g) = [π(x),π(x¯)] = e . That is, the
A
elementg ismappedtothetrivialelementinallamenablequotientsofG. Thus,Gisnotresidually
amenable. (cid:3)
8 FEDERICOBERLAI
In the next corollarywe construct countably many pairwise non-isomorphic groups, not known
to be sofic, satisfying Kaplansky’s direct and stable finiteness conjectures.
Corollary 4.2. With the notations of the previous theorem, for r ≥ 2 let F ≤Γ be generated by
r
{biab−i |i=0,...,r−1}. Then the groups
G :=Γ∗ Γ=hX,X¯ |R,R¯,a=a¯,...,br−1ab−r+1 =¯br−1a¯¯b−r+1i
r Fr
are pairwise non-isomorphic and are not LEA. Moreover they are free-by-sofic, and hence D[G ]
r
is directly finite (equivalently: stably finite) with respect to all division rings D, for all r ≥2.
Proof. ThesubgroupF isafreegroupofrankr. TheargumentoftheproofofTheorem4.1shows
r
thattheelementg =[x,x¯]ismappedtothetrivialelementinallamenablequotientsofG . Hence
r
G is not LEA.
r
By the universal property of amalgamated free products, we have the following commuting
diagram
Γ (cid:31)(cid:127) // G oo ?_ Γ¯
✷ r
✷ ☞
✷ ☞
✷ ☞
✷ ☞
✷ ☞
✷ ☞
✷ ∃!ϕ ☞
id ✷ ☞ id
✷ ☞
✷ ☞
✷ ☞
✷(cid:25)(cid:25) (cid:15)(cid:15) (cid:5)(cid:5)☞☞
Γ
where ϕ: G ։Γ is a surjective homomorphism. Let K =kerϕ, then K∩Γ=K∩Γ¯ ={e}. This
r
implies that K acts freely on the Bass-Serre tree associated to the amalgamated free product G ,
r
that is to say, K is a free group.
Hence G is free-by-soficand, givena division ring D, the groupring D[G ] is directly finite by
r r
our Main Theorem. Note that K is not finitely generated, so we cannot conclude that N[G ] is
r
directly finite for noetherian rings N.
Itremainstoprovethatthefamily{G } consistsofpairwisenon-isomorphicfinitelypresented
r r≥2
groups. To this aim, we recall the notion of deficiency of a finitely presented group.
The deficiency def(G) of a finitely presented group G is defined as max{|X|−|R|} over all the
finite presentations G=hX |Ri. It is invariant under isomorphism, and we have
def(G )=2·def(SL (Z[1/p])−r.
r m
Therefore, the groups {G } are pairwise non-isomorphic.
r r≥2
(cid:3)
Note that the groups G do not have property (T). This follows, for instance, from [6, Remark
r
2.3.5 and Theorem 2.3.6].
Our result extends further to HNN extensions. In [7], we have characterized the residual
amenability of particular HNN extensions A∗ and amalgamated free products A∗ A, in terms
H H
ofthe amalgamatedsubgroupH being closedin the proamenabletopology of A [7, Corollaries1.8
and 1.10]. Using these results and Corollary 4.2, we obtain:
Corollary 4.3. Let p > 2 be a prime number, n ≥ 3 and SL (Z[1/p]) = hX | Ri. For r ≥ 2 the
n
groups
Γ :=hX,t|R, tat−1 =a, t(bab−1)t−1 =bab−1,..., t(br−1ab−(r−1))t−1 =br−1ab−(r−1)i
r
are pairwise non-isomorphic and are not LEA. Moreover they are free-by-sofic, and hence D[Γ ] is
r
directly finite (equivalently: stably finite) with respect to all division rings D, for all r ≥2.
We end with the following question:
Question 4.4. Are the groups G and Γ sofic/hyperlinear?
r r
GROUPS SATISFYING KAPLANSKY’S STABLE FINITENESS CONJECTURE 9
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Universita¨t Wien,Fakulta¨t fu¨rMathematik,Oskar-Morgenstern-Platz1,1090Wien, Austria.
E-mail address: [email protected]